When parrots are held onto the shoulder, you're able to grab a maximum of two parrots before running out of room. However, attempting to pick up another tamed parrot whilst the other two are held on your shoulders will make it disappear. Parrots aren't replaced, but it surely looks like the parrot that was once in front of you is gone now.
Linked issues
is duplicated by 9
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Actually this happens in windows 7 too so its not a windows 10 only glitch